Diss / 10

then

which means, if the frequencies are detuned on the value multiple 2πT−1, the spectral components S1 (ω1) and S1 (ω2 ) are uncorrelated. When 0.5(ω1 − ω2)T 1, we can neglect the correlation function between the estimates of spectral density given by (15.238).

The variance of estimate of the function S(ω1) can be defined by (15.244) using the limiting case at ω1 = ω2 = ω:

Thus, according to (15.238), in spite of the fact that the estimate of spectral density ensures unbiasedness, it is not acceptable because the value of the estimate variance is larger than the squared spectral density true value.

Averaging the function S1 (ω) by a set of realizations is not possible, as a rule. Some indirect methods to average the function S1 (ω) are discussed in Refs. [3,5,6]. The first method is based on implementation of the spectral density averaged by frequency bandwidth instead of the estimate of spectral density defined at the point (the estimate of the given frequency). In doing so, the more the frequency range, within the limits of which the averaging is carried out, at T = const, the lesser the variance of estimate of spectral density. However, as a rule, there is an estimate bias that increases with an increase in the frequency range, within the limits of which the averaging is carried out. In general, this averaged estimate of spectral density can be presented in the following form:

where W(ω) is the even weight function of frequency ω or as it is called in other words the function of spectral window. The widely used functions W(ω) can be found in Refs. [3,5,6].

As T → ∞, the bias of spectral density estimate S2 (ω) can be presented in the following form:

is true, where S′(ω) and S″(ω) are the derivatives with respect to the frequency ω. Because of this, we can write

As we can see from (15.254), as T → ∞, Var{S2 (ω)} → 0; that is, the estimate of the spectral density S2 (ω) is consistent.

The second method to obtain the consistent estimate of spectral density is to divide the observation time interval [0, T] on N subintervals with duration T0 < T and to define the estimate S1i (ω) for each subinterval and subsequently to determine the averaged estimate

As we can see from (15.256), as T → ∞, the estimate given by (15.255) will be consistent.

In radar applications, sometimes it is worthwhile to obtain the current estimate S1 (ω, t) instead of the averaged summation given by (15.225). Subsequently, the obtained function of time is smoothed

by the low-pass filter with the filter constant time τfilter T0. This low-pass filter is equivalent to estimate by ν uncorrelated estimations of the function S1 (ω), where ν = τfilterT0−1. In practice, it makes sense to consider only the positive frequencies f = ω(2π)−1. Taking into consideration that the

correlation function and spectral density remain even, we can write

According to (15.228) and (15.229), the spectral density G(f) and the correlation function R(τ) take the following form:

In this case, the current estimate, by analogy with (15.238), can be presented in the following form:

where the squared current spectral density can be presented in the following form:

where Ac2 ( f , t) and As2 ( f , t) are the cosine and sine components of the current spectral density of realization x(t). As a result of smoothing the estimate G1 ( f , t) by the filter with the impulse response h(t), we can write the averaged estimate of spectral density in the following form:

The flowchart illustrating how to define the current estimate G2 ( f , t) of spectral density is shown in Figure 15.12. The input realization of stochastic process is multiplied by the sine and cosine signals of the reference generator in quadrature channels, correspondingly. Obtained products are integrated, squared, and come in at the summator input. The current estimate G2 ( f , t) forming at the summator output comes in at the smoothing filter input. The smoothing filter possesses the impulse response h(t). The smoothed estimate G(f, t) of spectral density is issued at the filter output.

568 Signal Processing in Radar Systems

Naturally, the accuracy of this approximation increases with concomitant decrease in the filter bandwidth f, since as f → 0 we can write

As applied to the ergodic stochastic processes, under definition of variance, the averaging by realization can be changed based on the averaging by time as T → ∞

For this reason, the value

is considered as the estimate of spectral density under designing and construction of measurers of stochastic process spectral density. The values f and 2max are known before. Because of this, a measurement of the stochastic process spectral density is reduced to estimate of stochastic process variance at the filter output. We need to note that (15.272) envisages a correctness of the condition T f  1, which means the observation time interval is much longer compared to the narrowband filter time constant.

Based on (15.272), we can design the flowchart of spectral density measurer shown in Figure 15.13. The spectral density value at the fixed frequency coincides accurately within the constant factor with the variance of stochastic process at the filter output with known bandwidth. Operation principles of the spectral density measurer are evident from Figure 15.13. To define the spectral density for all possible values of frequencies, we need to design the multichannel spectrum analyzer and the central frequency of narrowband filter must be changed discretely or continuously. As a rule, a shift by spectrum frequency of investigated stochastic process needs to be carried out using, for example, the linear law of frequency transformation instead of filter tuning by frequency. The structure of such measurer is depicted in Figure 15.14. The sawtooth generator controls the operation of measurer that changes the frequency of heterodyne.

FIGURE 15.13  Measurement of spectral density.

Heterodyne Generator Recoder

FIGURE 15.14  Measurement of spectral density by spectrum frequency shift.

Define the statistical characteristics for using the filter method to measure the spectral density of stochastic process according to (15.272). The mathematical expectation of spectral density estimate at the frequency f0 takes the following form:

In a general case, the estimate of spectral density will be biased, that is,

The variance of spectral density estimate is defined by the variation in the variance estimate of sto-

chastic process y(t, f0) at the filter output. If the condition T τcor is satisfied for the stochastic process y(t, f0), the variance of estimate is given by (13.64), where instead of S(ω) we should understand

To define the bias and variance of spectral density estimate of stochastic process we assume that the module of transfer function is approximated by the following form:

Thus, the bias of spectral density estimate of stochastic process is proportional to the squared bandwidth of narrowband filter. To define the variance of estimate for the first approximation, we can assume that the condition G(f) ≈ G(f0) is true within the limits of the narrowband filter bandwidth. Then, according to (15.276), we obtain

The dispersion of spectral density estimate of the stochastic process takes the following form:

15.6  ESTIMATE OF STOCHASTIC PROCESS SPIKE PARAMETERS

In many application problems we need to know the statistical parameters of stochastic process spike (see Figure 15.15a): the spike mean or the average number of down-up cross sections of some horizontal level M within the limits of the observation time interval [0, T], the average duration of the spike, and the average interval between the spikes. In Figure 15.15a, the variables τi and θi mean the random variables of spike duration and the interval between spikes, correspondingly. To measure these parameters of spikes, the stochastic process realization x(t) is transformed by the nonlinear transformer (threshold circuitry) into the pulse sequence normalized by the amplitude ητ with duration τi (Figure 15.15b) or normalized by the amplitude ηθ with duration θi (Figure 15.15c), correspondingly:

Using the pulse sequences ητ and ηθ, we can define the aforementioned parameters of stochastic process spike. Going forward, we assume that the investigated stochastic process is ergodic, as mentioned previously, and the following condition T τcor is satisfied.

FIGURE 15.15  Transformation of stochastic process realization x(t) into the pulse sequence: (a) Example of stochastic process spike; (b) Pulse sequence normalized by the amplitude ητ with duration Ti; (c) Pulse sequence normalized by the amplitude ηθ with duration θi.

For this reason, we use the approximations τav ≈ τ and θav ≈ θ.

The mathematical expectation of the average number of stochastic process spikes can be deter-

In other words, the estimate of average number of the stochastic process spikes at the level M within the limits of the observation time interval [0, T] is unbiased as a first approximation.

The estimate variance of the average number of stochastic process spikes at the level M can be presented in the following form:

In the case of ergodic stochastic processes, the average values can be written in the following form: